Value of Log Infinity

The log function also called a logarithm function which comes under most mathematical problems. The logarithmic function is used to reduce the complexity of the problems by reducing multiplication into addition operations and division into subtraction operations by using the properties of logarithmic functions. Here, the method to find the logarithm function is given with the help of the value of log infinity.

In general, the logarithm is classified into two types. They are

• Common Logarithmic Function
• Natural Logarithmic Function

The log function with base 10 is called the common logarithmic function and the log with base e is called the natural logarithmic function.

The logarithmic function is defined by,

Where x is the logarithm of a number ‘b’ and ‘a’ is the base of the log function that could be either replaced by the value ‘e’ or ‘10’. The variable ‘a’ should be any positive number and that should not be equal to 1.

What is Infinity?

In terms of log infinity log(y), as the number y increases infinitely, log(y) also increases infinitely even at a slower rate. The symbol used to denote infinity is “∞”

How to calculate the value of Log Infinity?

Now, let us discuss how to find the value of log infinity using the common log function and natural log function.

Value of log₁₀ infinity

The log function of infinity to the base 10 is denoted as “log₁₀ ∞” or “log ∞”

According to the definition of the logarithmic function, it is observed that

Base, a = 10 and 10^x = ∞

Therefore, the value of log infinity to the base 10 is as follows

Consider that 10^∞ = ∞, it becomes

As the value of b approaches infinity, the value of x also approaches infinity.

Log₁₀ infinity = ∞

Value of log_e infinity

The natural log function of infinity is denoted as “log_e ∞”. It is also known as the log function of infinity to the base e. The natural log of ∞ is also represented as ln( ∞)

Log_e ∞ = ∞ (or) ln( ∞)= ∞

Both the common logarithm and the natural logarithm value of infinity possess the same value.

Sample Problem

Question:

Evaluate the following limits

1. \(\begin{array}{l}\lim_{x\rightarrow \infty }e^{x}\end{array} \)
2. \(\begin{array}{l}\lim_{x\rightarrow -\infty }e^{x}\end{array} \)
3. \(\begin{array}{l}\lim_{x\rightarrow \infty }e^{-x}\end{array} \)
4. \(\begin{array}{l}\lim_{x\rightarrow -\infty }e^{-x}\end{array} \)

Solution:

(1)When x takes the value infinity, it becomes.

e^∞ = ∞

(2)When x takes the negative value infinity, it becomes.

e^–∞ = 0

(3)When x takes the value infinity, it becomes.

e^–∞ = 0

(4)When x takes the negative value of infinity, it becomes.

e^-(–∞)= ∞

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